Kähler-like characteristic connections
Abstract
For almost Hermitian manifolds in the Gray--Hervella class G1, characterised by the vanishing of the W2 component of the intrinsic torsion, we prove that the characteristic connection is K\"ahler-like with parallel Nijenhuis tensor if and only if its torsion is parallel and closed. In complex dimension three, the K\"ahler-like condition suffices. In the Hermitian case, this recovers a theorem of Zhao and Zheng. Our proof combines identities due to Ivanov and Stanchev with U(n)-representation theory, providing an alternative approach to the known characterisation. For Bismut K\"ahler-like metrics, the Lee form is Bismut-parallel and its dual is holomorphic and Killing; if it is nonzero, its complex flow defines a local Hermitian quotient with parallel Bismut torsion. Taking the product with S1 yields corresponding torsion characterisations for almost contact metric manifolds admitting a characteristic connection and local quotient results in the normal case. Finally, for almost quaternionic K\"ahler manifolds with torsion of real dimension at least eight, we prove that the torsion is parallel and closed if and only if the characteristic connection satisfies the Riemannian first Bianchi identity and the S3H-component of the torsion is parallel. We also show that the first Riemannian Bianchi identity alone implies that the structure is locally almost hyper-K\"ahler with torsion or quaternionic K\"ahler.
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BibTeX
@misc{marigliano2026kählerlike,
title = {{Kähler-like characteristic connections}},
author = {Luigi Marigliano},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05853346v1},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Kähler-like characteristic connections
AU - Luigi Marigliano
PY - 2026
DA - 2026/10/27
PB - HAL
UR - https://hal.science/hal-05853346v1
AB - For almost Hermitian manifolds in the Gray--Hervella class G_1, characterised by the vanishing of the W_2 component of the intrinsic torsion, we prove that the characteristic connection is K\"ahler-like with parallel Nijenhuis tensor if and only if its torsion is parallel and closed. In complex dimension three, the K\"ahler-like condition suffices. In the Hermitian case, this recovers a theorem of Zhao and Zheng. Our proof combines identities due to Ivanov and Stanchev with U(n)-representation theory, providing an alternative approach to the known characterisation. For Bismut K\"ahler-like metrics, the Lee form is Bismut-parallel and its dual is holomorphic and Killing; if it is nonzero, its complex flow defines a local Hermitian quotient with parallel Bismut torsion. Taking the product with S^1 yields corresponding torsion characterisations for almost contact metric manifolds admitting a characteristic connection and local quotient results in the normal case. Finally, for almost quaternionic K\"ahler manifolds with torsion of real dimension at least eight, we prove that the torsion is parallel and closed if and only if the characteristic connection satisfies the Riemannian first Bianchi identity and the S^3H-component of the torsion is parallel. We also show that the first Riemannian Bianchi identity alone implies that the structure is locally almost hyper-K\"ahler with torsion or quaternionic K\"ahler.
ER -